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We give a simple, randomized greedy algorithm for the maximum satisfiability problem (MAX SAT) that obtains a ¾ -approximation in expectation. In contrast to previously known ¾ -approximation algorithms, our algorithm does not use flows or linear programming. Hence we provide a positive answer to a question posed byWilliamson in 1998 on whether such an algorithm exists. Moreover, we show that Johnson's greedy algorithm cannot guarantee a ¾ -approximation, even if the variables are processed in a random order. Thereby we partially solve a problem posed by Chen, Friesen, and Zheng in 1999. In order to explore the limitations of the greedy paradigm, we use the model of priority algorithms of Borodin, Nielsen, and Rackoff. Since our greedy algorithm works in an online scenario where the variables arrive with their set of undecided clauses, we wonder if a better approximation ratio can be obtained by further fine-tuning its random decisions. For a particular information model we show that no priority algorithm can approximate Online MAX SAT within ¾ + ε (for any ε > 0). We further investigate the strength of deterministic greedy algorithms that may choose the variable ordering. Here we show that no adaptive priority algorithm can achieve approximation ratio ¾ . We propose two ways in which this inapproximability result can be bypassed. First we show that if our greedy algorithm is additionally given the variable assignments of an optimal solution to the canonical LP relaxation, then we can derandomize its decisions while preserving the overall approximation guarantee. Second we give a simple, deterministic algorithm that performs an additional pass over the input. We show that this 2-pass algorithm satisfies clauses with a total weight of at least ¾OPTLP, where OPTLP is the objective value of the canonical linear program. Moreover, we demonstrate that our analysis is tight and detail how each pass can be implemented in linear time. [ABSTRACT FROM AUTHOR]

We give a 7.18-approximation algorithm for the minimum latency problem that uses only O(n log n) calls to the prize-collecting Steiner tree (PCST) subroutine of Goemans and Williamson. This improves the previous best algorithms in both performance guarantee and running time. A previous algorithm of Goemans and Kleinberg for the minimum latency problem requires an approximation algorithm for the k-minimum spanning tree (k-MST) problem which is called as a black box for each value of k. Their algorithm can achieve an approximation factor of 10.77 while making O(n(n + log C) log n) PCST calls, a factor of 8.98 using O(n³ (n + log C) log n) PCST calls, or a factor of 7.18 + ϵ using nO(1/ϵ) log C PCST calls, via the k-MST algorithms of Garg, Arya and Ramesh, and Arora and Karakostas, respectively. Here n denotes the number of nodes in the instance, and C is the largest edge cost in the input. In all cases, the running time is dominated by the PCST calls. Since the PCST subroutine can be implemented to run in O(n²) time, the overall running time of our algorithm is O(n³ log n). We also give a faster randomized version of our algorithm that achieves the same approximation guarantee in expectation, but uses only O(log² n) PCST calls, and derandomize it to obtain a deterministic algorithm with factor 7.18 + ϵ, using O( 1/ϵ log² n) PCST calls. The basic idea for our improvement is that we do not treat the k-MST algorithm as a black box. This allows us to take advantage of some special situations in which the PCST subroutine delivers a 2-approximate k-MST. We are able to obtain the same approximation ratio that would be given by Goemans and Kleinberg if we had access to 2-approximate k-MSTs for all values of k, even though we have them only for some values of k that we are not able to specify in advance. We also extend our algorithm to a weighted version of the minimum latency problem. [ABSTRACT FROM AUTHOR]

We present a linear programming-based method for finding "gadgets," i.e., combinatorial structures reducing constraints of one optimization problem to constraints of another. A key step in this method is a simple observation which limits the search space to a finite one. Using this new method we present a number of new, computer-constructed gadgets for several different reductions. This method also answers a question posed by Bellare, Goldreich, and Sudan [SIAM J. Comput., 27 (1998), pp. 804-915] of how to prove the optimality of gadgets: linear programming duality gives such proofs. The new gadgets, when combined with recent results of Håstad [Proceedings of the 29th ACM Symposium on Theory of Computing, 1997, pp. 1-10], improve the known inapproximability results for MAX CUT and MAX DICUT, showing that approximating these problems to within factors of 16/17+ ∊ and 12/13+ ∊ , respectively, is NP-hard for every ∊ > 0. Prior to this work, the best-known inapproximability thresholds for both problems were 71/72 (M. Bellare, O. Goldreich, and M. Sudan [SIAM J. Comput., 27 (1998), pp. 804-915]). Without using the gadgets from this paper, the best possible hardness that would follow from Bellare, Goldreich, and Sudan and Håstad is 18/19. We also use the gadgets to obtain an improved approximation algorithm for MAX3 SAT which guarantees an approximation ratio of .801. [ABSTRACT FROM AUTHOR]